differential eqaution and linear algebra

Exam 2 - October 14, 2013

Instructions:

1. This test has 5 problems, totaling 100 points. Some problems have several parts.

2. No notes, books, or calculators.

3. Work and write clearly, and circle your final answers. If I cannot figure out your work/method because it’s unreadable or confusing, you will lose lots of points !

4. You will receive no credit, even for correct answers, without proper justifications. On the other hand, incorrect answers can receive substantial credit if your work demonstrates your understanding of the material. In short, help me give you partial credit by clearly showing all your work.

5. Manage your time carefully. Don’t spend too much time bogged down on any one problem. If you’re stuck, it’s o.k. Just move on to something else, and come back later.

6. Good luck !

Problem 1. (20 points, 10 minutes) By whatever method you wish, find the inverse of the matrix

A =

  1 2 00 1 2

0 0 1

  .

Problem 2. (20 points, 10 minutes) Given the differential equation E defined by:

E : y′(x) + 2e2x

1 + e2x y(x) = e−2x,

for every x ∈ R, find the general solution. You should give an explicit solution.

Problem 3. (20 points, 10 minutes) Study the linear system S : ax = b, where the matrices a,b are:

a =:

  0 −1 −1 20 −1 −1 0 −1 1 1 0

  , b =:

  10

0

  .

In particular, find:

a) the unknowns space of S;

b) the solution affine subspace sol(S) and its geometry;

c) dim(sol(S)) and one basis of sol(S);

d) the ranks ρ(a),ρ(b),ρ(a,b);

e) apply the Rouchè - Capelli theorem.

f) the solution linear subspace sol(S0) of the corresponding homogeneous system.

g) are rows of a linearly independent? Find the dimension of the row-space of a and one basis.

Problem 4. (20 points, 10 minutes)

(a). (10 points, 5 minutes) Prove that the set

V = {a sin + b cos : (a,b) ∈ R2}

is a subspace of the space C0(R) of all real continuous functions defined on R.

(b). (10 points, 5 minutes) Study the linear system S : ax = b, where the matrices a,b are:

a =:

( 1 2 0 0 1 0

) , b =:

( 0 0

) .

In particular, find:

a) the unknowns space of S;

b) the solution affine subspace sol(S) and its geometry;

c) dim(sol(S)) and one basis of sol(S);

d) the ranks ρ(a),ρ(b),ρ(a,b);

e) apply the Rouchè - Capelli theorem.

f) are rows of a linearly independent? Find the dimension of the row-space of a and one basis.

Problem 5. (20 points, 10 minutes) Three theoretical items are given below.

(a). (6 points, 3 minutes) Show that if A is an n×n matrix, then A−AT is a skew-symmetric matrix. [Suggestion: First write down what it means to say that X is a skew-symmetric matrix.]

(b). (8 points, 4 minutes) Give a specific example of matrices A and B for which the following statement is false:

rank(A + B) = rank(A) + rank(B).

(c). (6 points, 3 minutes) Show that if A is an (n,n) matrix that satisfies

A2 − 2A + In = 0n,

then A−1 = 2In −A.